Chem 332: Problem Set #4 Due in class: Friday, Feb. 15th

(1) Consider a quantum particle of mass m that is completely free to travel in one-dimension,

V(x) = 0.

(a) Write out the full expression for the time independent Schrödinger equation.

(b) Show that ψ (x) = Aeikx + Be−ikx is a general solution where k = 2mE2

.

(c) Consider the two cases where A=0 and then where B=0. Determine if these

wavefunctions are (separately) eigenfunctions of the momentum operator and, if so,

what the eigenvalues are.

(d) Are there any restrictions on the total energy for this particle?

(2) The function ψ x( ) = A xL

⎛⎝⎜

⎞⎠⎟2

1− xL

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

is an acceptable wave function for the particle in

a 1-dimensional infinitely deep box of length L.

a) Calculate the normalization constant A.

b) Calculate the average values of x and x2 for this state, and use these to calculate the

standard deviation, σ x = x2 − x 2 . Based on the Heisenberg uncertainty principle,

what then is the minimum standard deviation in the momentum σp?

(3) For a particle in a 1-dimensional infinite depth box of length L, the first excited state

wave function is ψ 2 x( ) = 2Lsin 2π

Lx⎛

⎝⎜⎞⎠⎟ . What is the probability that the particle will

be found in the middle third of the box?

(4) The eigenvalues of the particle in a box system form the basis for describing the

translational motion of atoms and molecules. For a single He atom in a 1-dimensional

box, calculate the value of the quantum number n of the energy level for which the

energy is equal to 32kBT at 300 K when the box is 1 nm long and also when it is 10-2 m

long. Under the latter conditions, what is the energy separation between quantum states

with n and n+1? What is the de Broglie wavelength?

(5) The energy states for a particle in a 3-dimensional box with lengths L1, L2, and L3 are

given by

E n1,n2 ,n3( ) = h2

8mn1

L1

⎛

⎝⎜⎞

⎠⎟

2

+n2

L2

⎛

⎝⎜⎞

⎠⎟

2

+n3

L3

⎛

⎝⎜⎞

⎠⎟

2⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

.

These energy levels are sometimes used to model the motion of electrons in a central metal

atom that is surrounded by six ligands.

a) Show that the lowest energy level is nondegenerate and the 2nd level is triply degenerate if

the box is cubical. Label the states by their quantum numbers n1, n2, n3.

b) Consider a box of volume V = L1L2L3 with 3 electrons inside (2 in the lowest energy level,

1 in the next). Show that the total energy in this case is equal to E =

h2

8m12L1

2

⎛

⎝⎜

⎞

⎠⎟ if

L1=L2=L3.

c) Compared to part (b), a lower total energy results upon rectangular distortion (L1=L2≠L3) at

constant volume (V), E =

h2

8m6

L12+

6L3

2

⎛

⎝⎜

⎞

⎠⎟ , where L3>L1 . Show that the ratio of L3 to

L1 that minimizes the total energy is equal to 2 . This problem is related to Jahn-Teller

distortions in molecules. (Hint: minimize E with respect to L1 and remember that

L3 =VL12 where V is constant.)

(6) Calculate the first 4 energy levels of the π-network in hexatriene, C6H8, using the free

electron molecular orbital model. To calculate the box length, assume that the molecule

is linear and use the values 135 and 154 pm for the C=C and C–C bonds, respectively.

Sketch and label an energy level diagram showing the occupied levels and the first

unoccupied one. What is the wavelength of light required to induce a transition from the

ground state to the first excited state? How does this compare with the experimentally

observed value of 240 nm?

Hexatriene: H2C=CH–CH=CH–CH=CH2